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In graph-theoretic mathematics, a star coloring of a graph G is a (proper) vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors has connected components that are star graphs. Star coloring has been introduced by Grünbaum (1973). The star chromatic number
One generalization of star coloring is the closely related concept of acyclic coloring, where it is required that every cycle uses at least three colors, so the two-color induced subgraphs are forests. If we denote the acyclic chromatic number of a graph G by
The star chromatic number has been proved to be bounded on every proper minor closed class by Nešetřil & Ossona de Mendez (2003). This results was further generalized by Nešetřil & Ossona de Mendez (2006) to all low-tree-depth colorings (standard coloring and star coloring being low-tree-depth colorings with respective parameter 1 and 2).
Complexity
It was demonstrated by Albertson et al. (2004) that it is NP-complete to determine whether